"If you want to win championships, those are the games that you've got to go get," Boucher said. "You've got to know how to do it."
Tampa Bay also got a goal from Simon Gagne. Mike Smith stopped 22 shots, helping the Lightning win for just the second time in their last 11 homes games against Buffalo.
"What can I say? These guys are just battling every night," Smith said.
Tampa Bay is 5-2-1 in its last eight games, including four one-goal wins.
"I think we're taking pride the last couple weeks in our defensive play," Stamkos said.
Jason Pominville had a goal and Ryan Miller made 34 saves for the Sabres. Miller was pulled after allowing four goals on 10 shots in the first period of a 6-2 loss Friday night at Florida.
"It's disappointing," Sabres right wing Mike Grier said of the consecutive losses. "We were playing fairly well coming in, but we dug ourselves a hole."
Lightning captain Vincent Lecavalier failed to score on a penalty shot with 5:45 remaining in the third.
Gagne put the Lightning up 1-0 when he lifted a low-slot backhander over Miller during a power play with 7:44 left in the second. Miller lost for the fourth time in 20 career decisions against Tampa Bay.
Stamkos made it 2-0 with 1:50 left in the second, scoring from the slot off a pass from St. Louis, who had received a pass through the slot from Ryan Malone.
Pominville cut it to 2-1 when he scored an in-close goal with 47.7 seconds to go in the second. The right wing has 13 goals and 29 points in 21 games against the Lightning.
"I just didn't think we won enough battles," Buffalo coach Lindy Ruff said.
Tampa Bay went up 3-1 on St. Louis' goal with 1.6 seconds to play. Stamkos assisted on the goal for his 21st assist this season.
Miller stopped Dana Tyrell's shot during a short-handed breakaway 4 minutes into second. The goalie also stopped another short-handed chance by Tyrell early in the third.
Tampa Bay outshot Buffalo 18-5, including five shots by Stamkos, during the second.
"The way we're winning is great," Lightning center Nate Thompson said. "We're playing from the start to the end. These tight games, we're getting contributions from everyone."
Buffalo C Tim Connolly broke his nose and went to the locker room after a check into the boards by Thompson 90 seconds into the game. Connolly returned for the start of the second period. "Badly fractured," Ruff said. ... Gagne has 13 goals and 30 points in 31 games against Buffalo. ... Sabres D Shaone Morrisonn (concussion) has resumed practicing with the team. He has missed six straight games.
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TAMPA, FL -DECEMBER 18: Ryan Malone #6 of the Tampa Bay Lightning stretches during pre-game skate against the...